San Jose captain Logan Couture felt he should have done more while players were fighting for the puck in front of the net.
"I won the face-off toward the middle, and somehow Rasmussen ended up with a partial breakaway out of it," he said. "I think Reimer stopped three shots from him, and then I had a chance to clear it, but I put it right onto Copp's stick."
Rasmussen had a goal and an assist, and Ville Husso made 28 saves for the Red Wings (20-18-8), who won for the second time in six games (2-3-1).
"I had [Rasmussen on a line] with [Dylan] Larkin and [Lucas] Raymond, but I put him back with Copp and [David Perron] because they've created some nice chemistry," Detroit coach Derek Lalonde said. "I rarely single a player out for an effort, especially on a team where we all need to play well to win games, but [Rasmussen] deserves it tonight."
Erik Karlsson had two assists, and Reimer made 32 saves for the Sharks (14-25-10), who have lost eight of 10 (2-5-3) and were coming off a 4-0 loss to the Boston Bruins on Sunday.
"I thought we were better tonight," San Jose coach David Quinn said. "It wasn't a great game, but it was solid. We had a lot more honesty, pace and compete in our game tonight."
The Red Wings took a 1-0 lead when Adam Erne tipped Dylan Larkin's point shot past Reimer at 2:17 of the second period. The assist was Larkin's 400th NHL point (162 goals, 238 assists) in his 549th game.